If CloudFlare serves a lot of traffic (i.e. people on the internet are requesting stuff from CloudFlare's servers), they get better peering agreements (i.e. pay less) from internet network providers.
When "normal" people/companies connect to the internet, they're paying for the connection. Regional ISPs likewise pay Tier 1 network providers (i.e. "global internet backbone") for the connection, and are charged by bandwidth. When "popular" companies connect to the internet, they don't pay - e.g. a lot of ISPs would host Netflix servers for free (that way, they avoid having to pay for Netflix traffic to Tier 1 providers, but can serve it locally instead).